Finite Element Models and Analysis Methods Used for Uncertainty Quantification

1.1 Parametric Finite Element Model

A code already developed at the Institute of Aircraft Design and Lightweight Struc­tures (IFL) is enhanced to generate a finite element model of the structure. It is based on the parametric description of airplane wing geometry and a layout of the load-bearing structure [1], [2]. The code is written in Patran Command Language (PCL) which allows an automated generation of finite element wing models by the preprocessor MSC Patrano.

A HIRENASD wind tunnel model [3] scaled up to 58 m of span is employed as a test structure for investigations carried out in the context of the MUNA project. The wing box structural layout and the arrangement of engines are taken on from the predecessing project [4] and resemble the wing of an AIRBUS A340 aircraft (see Fig. 1).

The geometric data are imported from an ASCII input file and are employed to generate a finite element shell model of the wing. A transonic transport aircraft design is used with the corresponding weights given in table 1 to evaluate the target lift for the calculation of aerodynamic and static inertial loads.

Due to a high number of required aeroelastic calculations, especially for the stochastic analysis presented in the second part of the work, the high order panel method HISSS is used instead of an Euler or RANS code to calculate the discrete aerodynamic nodal loads. The lack of accuracy when calculating a load distribution on the wing surface at higher Mach numbers had to be accepted so that the numeric costs could be kept reasonable. The finite-element solver NASTRAN was used to calculate the nodal displacements of the structural model.

The in-house code coupling library ifls [5] was employed to perform the fluid – structure interaction. The code handles the load and displacement transfer between non-conform grids by using a three-field approach in combination with Lagrange multipliers. The structure of the coupling routine allows the interaction between dif­ferent commercial numerical solvers. The converged angle of attack aEqSt of static aeroelastic equilibrium was estimated by ifls iteratively for given lift and flow con­ditions by variation of an overall (geometric) angle of attack ag of the wing.

Table 1 Weights for the transonic transport aircraft design used in this study Gross weight mTOW to 256 Fuselage and empennage unit structure + payload тЕр + m-N to 95 Wing structure mw to 35 Total fuel mass mp to 106 Propulsion group mpG to 20

Fig. 1 HIRENASD wing geometry and structural layout

Nodal loads calculated on the aerodynamic surface were transferred to the nodes of the structural grid by means of conservative interpolation in the area of the wing box. In the region of the flap and slat structure the aerodynamic nodal loads were applied to additionally created auxiliary structural nodes and tied to the wing box by multi point constraints of RBE3 type.

The static inertial loads including fuel weight, engine loads and the weight of the flap structure had also to be taken into account to represent realistic loading conditions. The flap and slat structure were idealized as point masses and connected to the spar structure by multi point constraints in the same way as the aerodynamic forces. The masses of the high lift devices were also required for this idealization and were estimated by handbook methods [6]. Tank loads were also modeled with point masses and RBE3s. The tank masses were evaluated for each wing bay by calculation of the volume taken by the fuel for a given degree of refueling.

The static inertial loads including fuel weight, engine loads and the weight of the flap structure had also to be taken into account to generate realistic load cases. The flap and slat structure were idealized as point masses and tied to the spar structure by multi point constraints in the same way as the aerodynamic forces. The masses of the high lift devices needed for this simplified approach were estimated by handbook methods [6]. Tank loads were also modeled with point masses and RBE3s. The tank mass was estimated for each wing bay by calculation of the volume taken by the fuel for a given degree of refuelling.

The wing box structure was sized with respect to strength criteria and constraints of buckling stability. Two load cases were selected for the sizing process: a 2,5g maneuver and the landing impact (see table 1). The strength sizing was carried out by a fully stressed design using stress distribution computed for limit loads and a yield-stress criterion. The design against buckling failure was performed by hand­book methods [7] using maximal allowable stresses for the compression panels

Table 2 Weights for the transonic transport aircraft design used in this study 2.5g maneuver Altitude H km 11 Mach number Ma – 0.82 Gross weight mrow to 256 landing impact Altitude H km 0 Mach number Ma – 0.2 Gross weight mrow to 182 Cruise flight (1g) Altitude H km 11 Mach number Ma – 0.82 Gross weight mrow to 256

as well as optimum design curves and semi-empirical formulas for estimation of stiffener spacing and cross-section geometry.

Due to constraints defining the highest permitted elastic deflection of the wing tip given in [4], the wing box was also sized under consideration of stiffness. For this additional sizing procedure the contribution of structural members to the wing deflection was calculated following the pattern of the modified fully utilized design method (MFUD) proposed by Patnaik et al [8]. For the constrained degree of free­dom (in this case it is the bending displacement) the sensitivity factors can be cal­culated for each component of the structure. These factors are defined as dw/dm where dw is a partial change of displacement and dm is a change of structural mass. The change of bending deformation and structural mass are evaluated by attach­ing additional material (by increasing wing thickness or stiffener cross-section) to each structural member and recalculating the displacement w of the modified struc­ture subjected to a reference load case. These sensitivity factors are used within the MFUD-procedure to weigh the increase of wall thickness of the structural members until the displacement constraint is achieved. This method permits to attach an addi­tional structural mass only in those areas of the wing box whose stiffness influences the given deformation at most. The weight of the structure sized with this approach was estimated to be very close to those obtained by a time-consuming optimization procedure [8].